This invention relates to classical least squares (CLS) multivariate spectral analysis methods. More particularly, this invention relates to an improvement to the prediction phase of the CLS method wherein the spectral shapes of constituents or other factors that were not measured in the calibration data set are added to the prediction data set prior to conducting the least squares prediction of at least one of the members of the measured calibration data set found in the prediction data set.
The basic descriptions of the standard CLS methods (also called linear unmixing models and K-matrix methods) are described in many references, see D. M. Haaland and R. G. Easterling, xe2x80x9cImproved Sensitivity of Infrared Spectroscopy by the Application of Least Squares Methods,xe2x80x9d Applied Spectroscopy 34, 539-548 (1980); D. M. Haaland and R. G. Easterling, xe2x80x9cApplication of New Least Squares Methods for the Quantitative Infrared Analysis of Multicomponent Samples,xe2x80x9d Applied Spectroscopy 36, 665-673 (1982); D. M. Haaland, R. G. Easterling, and D. A. Vopicka, xe2x80x9cMultivariate Least-Squares Methods Applied to the Quantitative Spectral Analysis of Multicomponent Samples,xe2x80x9d Applied Spectroscopy 39, 73-84 (1985); and D. M. Haaland, xe2x80x9cMethods to Include Beer""s Law Nonlinearities in Quantitative Spectral Analysis,xe2x80x9d in ASTM Special Technical Publication, Computerized Quantitative Infrared Analysis, G. L. McClure, Editor, STP 934, 78-94 (1987). CLS multivariate calibration methods have been hampered by the general restriction that the functional form of all sources of spectral variation in the calibration samples be known and included in the CLS analysis. This restriction has limited the widespread use of CLS methods relative to the less restrictive methods of partial least squares (PLS) and principal component regression (PCR) (see for example D. M. Haaland and E. V. Thomas, xe2x80x9cPartial Least-Squares Methods for Spectral Analyses 1: Relation to Other Multivariate Calibration Methods and the Extraction of Qualitative Information,xe2x80x9d Analytical Chemistry 60, 1193-1202 (1988)). From another reference by the inventor herein (D. M. Haaland, L. Han, and T. M. Niemczyk, xe2x80x9cUse of CLS to Understand PLS IR Calibration for Trace Detection of Organic Molecules in Water,xe2x80x9d Applied Spectroscopy 53, 390-395 (1999)) one may obtain a description of the standard classical least squares (CLS) calibration method. If one writes the linear additive equations (e.g., Beer""s law equations) in matrix form, the CLS model can be expressed as:
A=KC+EAxe2x80x83xe2x80x83Eq. (1)
where A is a pxc3x97n matrix of measured p absorbance intensities for n sample spectra (i.e., in this case the spectra are the columns of the spectral matrix), K is a pxc3x97m matrix of the m pure-component spectra of all spectrally active components in the samples, C is the mxc3x97n matrix of concentrations for the m components in the n samples, and EA is the pxc3x97n matrix of spectral errors in the model. The linear least squares solution for K for the model in Eq. 1 from a series of calibration samples with known component concentrations is
{circumflex over (K)}=ACT(CCT)xe2x88x921≈AC+xe2x80x83xe2x80x83Eq. (2)
where {circumflex over (K)} is the linear least squares estimate of the m pure-component spectra K and C+ is the pseudoinverse of C. A variety of methods, including singular value decomposition or QR decomposition (C. L. Lawson and R. J. Hanson, xe2x80x9cSolving Least Squares Problems,xe2x80x9d Prentice-Hall, Englewood Cliffs, N.J. (1974)) can be employed to improve the numerical precision of the solution to the pseudoinverse in Eq. (2). Therefore, the CLS calibration decomposes the spectral data into a set of basis vectors that are chemically relevant and readily interpretable. If all component concentrations are known for the calibration samples and these components are the only source of spectral change, then CLS yields least-squares estimates of the pure-component spectra as the components exist in the sample matrix over the concentration range of the calibration samples. These pure-component spectra will include linear approximations of interactions between molecules if they are present in the calibration spectra. The CLS calibration model has generally included only spectrally active chemical components. However, it is also possible to include other sources of spectral variation present in the spectra that are not due to changes in the chemical compositions of the samples. Examples of additional sources of spectral variation that might be added to the CLS calibration include changes in purge gas H2O and CO2 concentrations, spectral changes in the sample spectra that represent spectrometer drift, and optical insertion effects. A Beer""s law drift component can be included in the CLS calibration if the drift is linearly related to some parameter that can be measured.
The CLS prediction uses the same model in Eq. 1 that was used for CLS calibration. The least-squares solution for the component concentrations of one or more sample spectra is given by
Ĉ=({circumflex over (K)}T{circumflex over (K)})xe2x88x921{circumflex over (K)}TA≈{circumflex over (K)}+Axe2x80x83xe2x80x83Eq. (3)
where now A represents the spectral matrix of the samples to be predicted and {circumflex over (K)}+ represents the pseudoinverse of {circumflex over (K)}.
A method for performing an improved classical least squares multivariate estimation of the quantity of at least one constituent of a sample comprising first utilizing a previously constructed calibration data set expressed as matrix {circumflex over (K)} representing the combination of vectors expressing the spectral shapes and concentrations of the measured pure sample constituents of the calibration data set, measuring the response of the sample that contains the constituents in the calibration data set as well as additional constituents and additional system effects not present in the calibration data set to form a prediction data set, adding at least one vector expressing the spectral shape (but not concentration) of at least one additional constituent or additional system effect not present in the calibration data set but present in the prediction data set to form an augmented matrix {circumflex over ({tilde over (K)})}, and estimating the quantity of at least one of the constituents in the calibration data set that is present in the sample by utilizing the augmented matrix {circumflex over ({tilde over (K)})}.
There is a wide variety of constituents and system effects that can be added during the prediction (estimation) phase of the classical least squares (CLS) analysis. Chemicals that were omitted from the calibration data set can be added as spectral shapes to form the augmented matrix. Other candidates for admission include but are not limited to spectrometer drift, temperature effects, optical changes due to sample insertion variations, shifts between spectrometers, differences in the chemicals in the calibration data set and their very close analogs present in the sample, chromatic aberrations, diffraction effects and nonlinearities present in the calibration samples. Addition of spectral shapes for baseline corrections has been known for some years now and is not part of this invention.